Lower bounds of Dirichlet eigenvalues for a class of higher order degenerate elliptic operators

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Abstract

Let \(\Omega \) be a bounded open domain in \({\mathbb {R}}^{n}\) with smooth boundary \(\partial \Omega \), \(X=(X_{1},X_{2},\ldots ,X_{m})\) be a system of real smooth vector fields defined on \(\Omega \) and the boundary \(\partial \Omega \) is non-characteristic for X. Denote \(\lambda _{k}\) as the k-th Dirichlet eigenvalue for degenerate elliptic operator L on \(\Omega \) with \(L=\left( \sum _{j=1}^mX_j^{2p}\right) ^{2}\), \(p\ge 1\), then in this paper, we give a lower bound estimate of \(\lambda _k\) for the operator L by using weighted Sobolev embedding theorem and maximally hypoelliptic estimate.

Keywords

Eigenvalues Maximally hypoelliptic estimate Hörmander’s condition Métivier’s condition 

Mathematics Subject Classification

35J70 35P15 

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Hua Chen
    • 1
  • Hongge Chen
    • 1
  • Junfang Wang
    • 1
  • Nana Zhang
    • 1
  1. 1.School of Mathematics and StatisticsWuhan UniversityWuhanChina

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